# Full text of "Analytical Mechanics"

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```IMPULSE AND MOMENTUM
245
common velocity. Then the deformed parts of the spheres begin to regain, at least partially, their original forms and cause the spheres to separate.
The process of collision may, therefore, be divided into two parts. The first part lasts from the initial contact at t — 0 until the instant when the centers of the spheres are nearest together at t = ti. The second part begins at t = ti and lasts until the spheres separate at t = &'. The impulse imparted to each body during the first part of the collision is called the impulse of compression, while that im-parted during the second part is called the impulse of restitution.
Let mi and m2 be the masses of the colliding bodies, vi and
v2 be their velocities just before and vi' and v2' just after the collision, and let v be their common velocity at the instant of maximum compression, that is, when the distance between the centers of mass is shortest. Further, let L ancf L' denote the impulses of compression and of restitution, respectively. Then we have
L = /  lFdt = mi (w — Vi) = — ra2 (v — tfc),
Jo
— v).
118.
f = / J ti
dt =
i — v) = — w2
The foregoing relations follow at once from the definition of impulse and from the fact that the colliding bodies form a system which is not acted upon by external forces, and consequently the sum of their momenta remains constant during the collision.```